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29
Recursive Circulants and their Embeddings among Hypercubes
"... We propose an interconnection structure for multicomputer networks, called recursive circulant. Recursive circulant G(N; d) is dened to be a circulant graph with N nodes and jumps of powers of d. G(N; d) is node symmetric, and has some strong hamiltonian properties. G(N; d) has a recursive structure ..."
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Cited by 17 (10 self)
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We propose an interconnection structure for multicomputer networks, called recursive circulant. Recursive circulant G(N; d) is dened to be a circulant graph with N nodes and jumps of powers of d. G(N; d) is node symmetric, and has some strong hamiltonian properties. G(N; d) has a recursive structure when N = cd m , 1 c < d. We develop a shortest path routing algorithm in G(cd m ; d), and analyze various network metrics of G(cd m ; d) such as connectivity, diameter, mean internode distance, and visit ratio. G(2 m ; 4), whose degree is m, compares favorably to the hypercube Qm . G(2 m ; 4) has the maximum possible connectivity, and its diameter is d(3m 1)=4e. Recursive circulants have interesting relationship with hypercubes in terms of embedding. We present expansion one embeddings among recursive circulants and hypercubes, and analyze the costs associated with each embedding. The earlier version of this paper appeared in [21].
Hypercube Embedding Heuristics: An evaluation
, 1990
"... The hypercube embedding problem, a restricted version of the general mapping problem, is the problem of mapping a set of communicating processes to a hypercube multiprocessor. The goal is to find a mapping that minimizes the length of the paths between communicating processes. Unfortunately the hype ..."
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Cited by 10 (3 self)
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The hypercube embedding problem, a restricted version of the general mapping problem, is the problem of mapping a set of communicating processes to a hypercube multiprocessor. The goal is to find a mapping that minimizes the length of the paths between communicating processes. Unfortunately the hypercube embedding problem bas been shown to be NPhard. Thus many heuristics have been proposed for hypercube embedding. This paper evaluates several hypercube embedding heuristics, including simulated annealing, local search, greedy, and recursive mincut bipartitioning. In addition to known heuristics, we propose a new greedy heuristic, a new KernighanLin style heuristic, and some new features to enhance local search. We then assess variations of these strategies (e.g., different neighborhood structures) and combinations of them (e.g., greedy as a front end of iterative improvement heuristics). The asymptotic running times of the heuristics are given, based on efficient implementations using a priorityqueue data structure.
On Embedding Binary Trees into Hypercubes
, 1997
"... Hypercubes are known to be able to simulate other structures such as grids and binary trees. It has been shown that an arbitrary binary tree can be embedded into a hypercube with constant expansion and constant dilation. This paper presents a simple lineartime heuristic which embeds an arbitrary b ..."
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Cited by 9 (1 self)
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Hypercubes are known to be able to simulate other structures such as grids and binary trees. It has been shown that an arbitrary binary tree can be embedded into a hypercube with constant expansion and constant dilation. This paper presents a simple lineartime heuristic which embeds an arbitrary binary tree into a hypercube with expansion 1 and average dilation no more than 2. We also give some results extending good embeddings for paritybalanced binary trees to arbitrary binary trees. In particular, we show that a conjecture of Havel [8] implies embeddings of binary trees into hypercubes with expansion 1 and either dilation 2 or average dilation approaching 1, and embeddings with expansion 2 and dilation 1.
Embedding Large Complete Binary Trees in Hypercubes with Load Balancing
, 1996
"... This paper presents two methods for embedding arbitrarily large complete binary trees in fixed size hypercubes. The ability to embed arbitrarily large graphs in smaller graphs has important applications in massively parallel computing. The presented embedding methods are optimized mainly for balanci ..."
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Cited by 5 (1 self)
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This paper presents two methods for embedding arbitrarily large complete binary trees in fixed size hypercubes. The ability to embed arbitrarily large graphs in smaller graphs has important applications in massively parallel computing. The presented embedding methods are optimized mainly for balancing the processor loads, while minimizing dilation and congestion as far as possible.
Optimal simulation of full binary trees on faulty hypercubes
 IEEE Transaction on Parallel and Distributed Systems
, 1995
"... Abst We study the problem of running full binary tree based algorithms on a hypercube with faulty nodes. The key to this problem is to devise a method for embedding a full binary tree into the faulty hypercube. Based on a novel embedding strategy, we present two results for embedding an (n 1)tre ..."
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Abst We study the problem of running full binary tree based algorithms on a hypercube with faulty nodes. The key to this problem is to devise a method for embedding a full binary tree into the faulty hypercube. Based on a novel embedding strategy, we present two results for embedding an (n 1)tree (a full binary tree with 2"l 1 nodes) into an ncube (a hypercube with 2 " nodes) with unit dilation and load. For the problem where the root of the tree must be mapped to a specified hypercube node (spec@d root embedding problem), we show that up to n 2 (node or edge) faults can be tolerated. This result is optimal in the following sense: 1) it is timeoptimal, 2) (n 1)tree is the largest full binary tree that can be embedded in an ncube, and 3) n 2 faults is the maximum number of worstcase faults that can be tolerated in the specNed root problem. Furthermore, we also show that any algorithm for this problem cannot be totally recursive in nature. For the problem where the root cm be mapped to any anonfaulty hypercube node (variable root embedding problem), we show that up to 2n 3 [log n1 faults can be tolerated. Thus we have improved upon the previous result of n 1 pog nl. In addition, we show that the algorithm for the variable root embedding problem is optimal within a class of algorithms d e d recursive embedding algorithms as far as the number of tolerable faults is concerned. Finally, we show that when an O(l/,/K) fraction of nodes in the hypercube are faulty, it is not always possible to have an O(1)load variable root embedding no matter how large the ~ dilation is. Index TermsEmbedding, hypercubes, full binary trees, dilation, simulation, faulty architecture. 1.
A General Method for Efficient Embeddings of Graphs into Optimal Hypercubes
 Proc. of the 2nd International EuroPar Conference on Parallel Processing, Vol. I, LNCS 1123
, 1996
"... Embeddings of several graph classes into hypercubes have been widely studied. Unfortunately, almost all investigated graph classes are regular graphs such as meshes, complete trees, pyramids. In this paper, we present a general method for onetoone embedding irregular graphs into their optimal hype ..."
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Embeddings of several graph classes into hypercubes have been widely studied. Unfortunately, almost all investigated graph classes are regular graphs such as meshes, complete trees, pyramids. In this paper, we present a general method for onetoone embedding irregular graphs into their optimal hypercubes based on extendededgebisectors of graphs. An extendededgebisector is an edgebisector with the additional property that a subset of the vertices is distributed more or less evenly among the two halves of the bisected graph. The dilation and congestion of the embedding depends on the quality of the extendededgebisector. Moreover, if the...
Optimal dynamic embeddings of complete binary trees into hypercubes
 J. Parallel Distrib. Comput
, 2001
"... It is folklore that the doublerooted complete binary tree is a spanning tree of the hypercube of the same size. Unfortunately, the usual construction of an embedding of a doublerooted complete binary tree into a hypercube does not provide any hint on how this embedding can be extended if each leaf ..."
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It is folklore that the doublerooted complete binary tree is a spanning tree of the hypercube of the same size. Unfortunately, the usual construction of an embedding of a doublerooted complete binary tree into a hypercube does not provide any hint on how this embedding can be extended if each leaf spawns two new leaves. In this paper, we present simple dynamic embeddings of doublerooted complete binary trees into hypercubes which do not suffer from this disadvantage. We also present edgedisjoint embeddings with optimal load and unit dilation. Furthermore, all these embeddings can be efficiently implemented on the hypercube itself such that the embedding of each new level of leaves can be computed in constant time. Because of the similarity, our results can be immediately transfered to complete binary trees. 2001 Academic Press Key Words: dynamic embedding; complete binary tree; hypercube; simulation of algorithms.
ProductClosed Networks
"... We present a uniform mathematical characterization of interconnection network classes referred to as productclosed networks (PCN). A number of popular network classes fall under this characterization including binary hypercubes, tori, kary ncubes, meshes, and generalized hypercubes. An unlimited ..."
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We present a uniform mathematical characterization of interconnection network classes referred to as productclosed networks (PCN). A number of popular network classes fall under this characterization including binary hypercubes, tori, kary ncubes, meshes, and generalized hypercubes. An unlimited number of other networks can be defined using the presented mathematical characterization. An important common feature for all PCN classes is their closure under the Cartesian product of graphs. This provides a tool for generating new PCN classes of interconnection graphs. We evaluate a number of commonly used metrics for all PCN networks including the size, degree, diameter, average distance, connectivity, and fault diameter. We show how all PCN networks share various desirable properties such as simple distributed routing, hierarchical structure, complete sets of nodedisjoint paths between arbitrary nodes, attractive embeddings, distributed broadcasting, and fault tolerance properties. Th...
On Embedding Various Networks Into The Hypercube Using Matrix Transformations
 IEEE Parallel Processing Symposium
, 1996
"... Various researchers have shown that the binary n cube (or hypercube) can embed any rary mcubes, having the same number of nodes, with dilation 1. Their construction method is primarily based on the reflected Gray code. We present a different embedding method based on matrix transformations&apos ..."
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Various researchers have shown that the binary n cube (or hypercube) can embed any rary mcubes, having the same number of nodes, with dilation 1. Their construction method is primarily based on the reflected Gray code. We present a different embedding method based on matrix transformations' schemes that achieves the same results. In addition, this method has a nice property that makes it suitable to be used in divideandconquer algorithms. Thus, it constitutes a useful tool for the design of parallel algorithms for the hypercube. 1 Introduction The embedding of a guest graph G into a host graph H is an injection (onetoone mapping) of the nodes in G to the nodes in H. the dilation cost of an embedding of G in H is the maximum distance in H between the images of any two neighboring nodes in G. This cost gives a measure of the proximity in H of the neighboring nodes in G under an embedding. Graph embedding results have many important applications in parallel processing. They ...
Efficient Dynamic Embeddings of Binary Trees into Hypercubes
 Journal of Algorithms
"... In this paper, a deterministic algorithm for dynamically embedding binary trees into hypercubes is presented. Because of a known lower bound, any such algorithm must use either randomization or migration, i.e., remapping of tree vertices, to obtain an embedding of trees into hypercubes with small di ..."
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In this paper, a deterministic algorithm for dynamically embedding binary trees into hypercubes is presented. Because of a known lower bound, any such algorithm must use either randomization or migration, i.e., remapping of tree vertices, to obtain an embedding of trees into hypercubes with small dilation, load, and expansion simultaneously. Using migration of previously mapped tree vertices, the presented algorithm constructs a dynamic embedding which achieves dilation of at most 9, unit load, nearly optimal expansion, and constant edge and nodecongestion. This is the first dynamic embedding that achieves these bounds simultaneously. Moreover, the embedding can be computed efficiently on the hypercube itself. The amortized time for each spawning step is bounded by O�log 2 �L��, if in each step at most L new leaves are spawned. From this construction, a dynamic embedding of large binary trees into hypercubes is derived which achieves dilation of at most 6 and nearly optimal load. Similarly, this embedding can be constructed with nearly optimal load ρ on the hypercube itself in amortized time O�ρ · log 2 �L/ρ�� per spawning step, if in each step at most L new leaves are added. © 2002 Elsevier Science (USA) Key Words: dynamic embedding; binary tree; hypercube; simulation of algorithms.